Question

$S_1$ : Sum of 5 consecutive odd numbers
$S_2$ : Sum of 4 consecutive even numbers

Sum of twice of $S_1$ and thrice of $S_2$ is 178. Also, the sum of thrice of $S_1$ and twice of $S_2$ is 177.

If all the 9 numbers are arranged in ascending order, then find the sum of the smallest and largest number.

Answer 1

Let the odd numbers be $x,x+2,x+4,x+6andx+8.$
Sum, $S_1$ = $5x+20$

Let the even numbers be $y,y+2,y+4andy+6.$
Sum, $S_2$ = $4y+12$

$\Rightarrow$ $2(5x+20)+3(4y+12)=178$ and
$3(5x+20)+2(4y+12)=177$

On simplifying these two equations, we get

$5x+6y=51$ ...(1)
$15x+8y=93$ ...(2)

On multiplying equation (1) by 3, we get
$15x+18y=153$ ...(3)

On subtracting equation (2) from (3), we get
$10y=60\Rightarrow y=6$
On substituting $y=6$ in equation (1), we get $x=3.$

Therefore, the odd numbers are 3, 5, 7, 9, 11 and the even numbers are 6, 8, 10, 12.

Arranging them in ascending order,
the numbers are 3,5,6,7,8,9,10,11,12.

Hence the required sum is 3 + 12 = 15.